reserve A,B for limit_ordinal infinite Ordinal;
reserve B1,B2,B3,B5,B6,D, C for Ordinal;
reserve X for set;
reserve X for Subset of A;

theorem Th4:
  X is bounded iff ex B1 st B1 in A & X c= B1
proof
A1: sup X c< A iff sup X c= A & sup X <> A by XBOOLE_0:def 8;
A2: X c= sup X by Th2;
  A = sup A by ORDINAL2:18;
  then sup X <> A iff sup X in A by A1,ORDINAL1:11,ORDINAL2:22;
  hence X is bounded implies ex B1 st B1 in A & X c= B1 by A2;
  given B1 such that
A3: B1 in A and
A4: X c= B1;
  sup X c= sup B1 by A4,ORDINAL2:22;
  then sup X c= B1 by ORDINAL2:18;
  hence thesis by A3,ORDINAL1:12;
end;
